Difference between revisions of "2014 AIME II Problems/Problem 5"
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− | Let r, s, -r-s be the roots of p(x) (per Vieta's). Then <math>r^3 + ar + b = 0</math> and similarly for s. Also, <cmath>q(r+4) = (r+4)^3 + a(r+4) + b = 12r^2 + 48r + | + | Let r, s, -r-s be the roots of p(x) (per Vieta's). Then <math>r^3 + ar + b = 0</math> and similarly for s. Also, <cmath>q(r+4) = (r+4)^3 + a(r+4) + b + 240 = 12r^2 + 48r + 304 + 4a = 0</cmath> Set up a similar equation for s: <cmath>q(s-3) = (s-3)^3 + a(s-3) + b + 240 = -9s^2 + 27s + 213 - 3a = 0</cmath>. Simplifying and adding the equations gives <cmath>3r^2 - 3s^2 + 12r + 9s + 147 = 0 (*)</cmath> Now, let's deal with the a*x. Equating the a in both equations (per Vieta) <cmath>rs + (-r-s)(r+s) = (r+4)(s-3) + (-r-s-1)(r+s+1)</cmath>, which eventually simplifies to <cmath>s = \frac{13 + 5r}{2}</cmath> Substitution into (*) should give r = -5 and r = 1, corresponding to s = -6 and s = 9, and |b| = 330 and 90, for an answer of <math>\boxed{420}</math>. |
Revision as of 11:40, 31 March 2014
Problem 5
Real numbers and are roots of , and and are roots of . Find the sum of all possible values of .
Solution
Let r, s, -r-s be the roots of p(x) (per Vieta's). Then and similarly for s. Also, Set up a similar equation for s: . Simplifying and adding the equations gives Now, let's deal with the a*x. Equating the a in both equations (per Vieta) , which eventually simplifies to Substitution into (*) should give r = -5 and r = 1, corresponding to s = -6 and s = 9, and |b| = 330 and 90, for an answer of .